Mathematicians love to tell stories. Gauss’s youthful successes, the tragedy of Galois, Wiener’s absent-mindedness, Dantzig’s successful solution of the “homework problems” he found on the board: we tell the stories in class and to each other, perhaps because it is delightful to know that not all mathematicians are quite as boring as we are. Like most folk tales, the stories grow and change in the telling. Sometimes they migrate from one mathematician to another. At other times they just acquire refinements and decorations.

The whole thing drives historians nuts, of course. Many of the stories we tell are simply not true. Others, derived from a sort of oral tradition, cannot be verified. Most have the particularly annoying character of being partly true, making the historian’s objections seem like mere nitpicking.

Perhaps, however, we can learn something from those stories. They certainly are not reliable evidence for what really happened, but they might give evidence of how mathematicians think (or thought) about mathematics. Does the tragic account of poor misunderstood Galois suggest that mathematicians view (or viewed) themselves as tragic heroes? Amir Alexander thinks so: an analysis of our shared stories is at the center of Duel at Dawn.

The central thesis of the book is that in the early nineteenth century there was a change in how mathematicians viewed themselves. The “ideal mathematician” of the Enlightenment, Alexander argues, was a “child of nature,” someone who had not lost the natural innocence and clear perception of childhood and who was therefore able to penetrate deeply into Nature’s secrets. Mathematicians of the time insisted that their work was about the real world, that the mathematical patterns they found in nature were really there and not imposed from the outside. They did not worry too much about rigorous proof, because they felt that real phenomena could be relied on to distinguish true from false results. If solid proof was available, good, but if not, agreement with experiment would do.

Alexander argues that this changed in the 1820s and 1830s, with the story of Galois being the central example of the new point of view. He is careful to tell us the real story, demonstrating just how fanciful the “standard account” of Galois is — in fact, he shows that much of the “misunderstood genius” story was created by Galois himself and propagated by his friends, partly for political reasons. He notes that quite a similar story was told about Abel (whose real story is quite unromantic). Even Cauchy, the “establishment villain” in Galois’ story, thought of himself as misunderstood and persecuted for his devotion to the truth. This adds up to a strong argument that such stories reflect how mathematicians would like to be viewed.

What do the new stories tell us? They indicate, says Alexander, that the “ideal mathematician” of the nineteenth century was quite different from the Enlightenment picture. He did not quite fit into the mundane world. He was in contact with a deeper and truer reality from whence came his mathematical ideas. As a result, he was misunderstood and perhaps persecuted. He was fundamentally a tragic figure, whose mathematics was snatched out of the fire, so to speak. He accomplished much but died early, leaving us to wonder what else might have been known had he lived further.

This new story correlates, according to Alexander, with a new view of what mathematics is, one stressing universal truths and not the connection to reality. For the new generation of Romantic mathematicians, rigorous proof was the only guarantee of mathematical correctness because mathematics dealt with a higher reality that could not be tested experimentally. It was pursued not for its utility but for “the glory of the human spirit” (a phrase due to Jacobi). Otherworldly mathematicians were the ideal explorers of an otherworldly mathematics.

Of course, both of these views about mathematics match well with the artistic and literary culture of their times. The idea that children are naturally innocent and particularly open to the world comes from Rousseau and was a big part of the Enlightenment ethos. Similarly, tragic figures such as Keats, Byron, and young Werther were central in the Romantic movement of the early nineteenth century. Part of Alexander’s goal, then, is to portray mathematics as a cultural product affected by societal trends and fashionable ideas.

Alexander does an excellent job of setting out and arguing for his thesis. Perhaps the best sign of this is that the reader wants to argue back. How well-founded is all this?

My first hesitation has to do with the evidence for what Alexander describes as the Enlightenment view. Most of the examples he presents come from France and in particular from Condorcet’s obituaries. (As secretary of the Académie, producing such obituaries was one of Condorcet’s duties.) It seems beyond doubt that Condorcet was deeply influenced by Rousseau and that this is reflected in his account of d’Alembert, who was a personal friend. It is also visible, though perhaps less clearly, in his éloge of Euler. But one wonders what would have been said by others. In particular, one wonders whether anyone outside France really considered d’Alembert as prominent a figure as Alexander makes him.

One also worries whether potential counterexamples have been weighed as seriously as they should. While the stories of Cauchy, Galois, Abel, and Bolyai can easily be cast into the “tragic hero” mold, that is much harder to do with Gauss, Kummer, or Liouville. In fact, the treatment of Gauss in Duel at Dawn sometimes made me think of a tour guide insisting “let’s move on, nothing to see here…”

Another sort of counterexample can be found in the work of Green and Riemann. The life stories of both can easily be told in the Romantic style, but their work does not show the insistence on rigor and lack of connection to natural phenomena that Alexander describes as characteristic of Romantic mathematics. Riemann famously offered, in one of his papers, a physical argument for the solvability of a boundary value problem: since a charge distribution on this surface does produce an electric field inside it, there must exist a function describing that field, hence a solution to the relevant equation. This is precisely the kind of “natural mathematics” that Alexander associates to the Enlightenment!

I am also hesitant about Alexander’s claim that the stories from the Romantic period are still the dominant stories and that the view of mathematics associated with them is still the dominant view. One would like to hear further discussion about that. Weierstrass, for example, does not strike me as at all “Romantic” (more like Parnassian, I’d say). I am not persuaded that Grothendieck, which Alexander mentions often, fits the mold either.

Jeremy Gray has argued, in his Plato’s Ghost, that the mathematics of the late nineteenth and early twentieth centuries is best described as Modernist. Someone should bring Gray and Alexander together for a debate!

Finally, I wonder whether the method itself does not need to be used with a little more care. I do believe that the stories we tell can reveal things about us, but Alexander’s readings of these stories sometimes seem a bit too direct and unsubtle. He assumes a bit too easily that we want to be like the heroes of our stories. I wonder, for example, how many mathematicians ever thought of Galois as in any sense the “ideal mathematician.” We certainly retell the story (as put forth by his friends and biographers, and especially by E. T. Bell), but when we do, are we saying that this is what doing mathematics should be like?

I wonder whether what makes these stories interesting is not precisely how different their protagonists are from the typical mathematician. A feature that pervades many of the stories we tell is simply the emphasis on precociousness. The early deaths of Abel, Galois, Riemann, and Eisenstein highlight the amazing achievements of their youth, as do some of the stories about Gauss. This aspect of the stories does not appear at all in Alexander’s book, but this theme seems much more prevalent than that of the misunderstood genius.

Or is it simply that these are good stories? Every once in a while, in fact, Alexander himself seems to be lured into the delight of stories, losing his historical objectivity. The clearest example is when Gauss is cast as the establishment villain in Alexander’s account of Bolyai — surely it follows from his own argument that one must be suspicious of such stereotyped narratives.

Alexander’s overall thesis is interesting and provocative; historians will want to look further, of course. Perhaps the best tribute to a serious historical thesis is to argue with it!

Hesitations aside, there is a lot of good stuff in the book. If there really are people who think of mathematics as supra-historical and otherworldly, they will find Alexander’s insights challenging and illuminating. The analysis of the stories of d’Alembert, Abel, Galois, and Cauchy is very well done, sorting out true from false and tracking down how the received account developed. Alexander clearly loves telling stories and he has a point to make, giving the reader both the pleasure of well-told tales and the opportunity for intellectual engagement. Most of all, Duel at Dawn gives readers the opportunity to engage with the history of mathematics at a level that is deeper than mere anecdotes. It deserves many readers.

Fernando Q. Gouvêa often has to explain to his students at Colby College that the History of Mathematics course does not consist of “stories about mathematicians.”